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**WAVE MOTION**

**Beats**

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__Description__

Beats is a phenomenon associated with sound waves, though the effect applies to all waves.

Essentially, when two similar frequencies (** f _{1}** ,

**f**) are sounded, a third much lower frequency is heard at the same time.

_{2}

This third frequency is called the **beat frequency** (** f_{B }**).

The beat frequency is simply the difference between the two original frequencies.

The beat frequency is measured from the rise and fall in the loudness/volume.

There is yet another frequency called the **combined frequency**(** f_{C}**).

This is the result of superposition of the two original frequencies. The combined frequency is simply the average of these frequencies.

Since the frequencies ** f _{1}** ,

**f**are almost the same, the change in frequency to

_{2}**is hardly noticeable.**

*f*_{C}

An example of the effect is the sound from a twin engined prop. aircraft. There is a periodic '**wow**' or '**drone**' noise produced as a result of the change in r.p.m. from the different propeller blades.

__Explanation__

The effect is a result of superposition of two sound wave frequencies producing a succession of constructive and distructive interference.

When the two frequencies are in phase they add, producing a wave with double the amplitude.

When the two waves are out of phase, they destroy eachother.

__Theory__

Consider our two original frequencies **f _{1}** and

**f**.

_{2}

In time * t *the number of cycles completed by each frequency is

**f**and

_{1}t*.*

**f**_{2}t

**no. cycles = (no cyles per second) x (no. seconds**)

Let us choose the time * t* such that the first wave completes one more cycle than the second.

From the first of two images (above), * t* is the time interval between the waves being in phase with each other.

So * t* is the

**beat period**

*(time for one complete 'beat' wave).*

**T**

For any wave, period and frequency are inversely proportional to one another.

So for beat period * T* and beat frequency

**,**

*f*_{B}

hence, by similarity between the last two equations,

assuming **f _{1}** >

**f**

_{2}

__Measuring an unknown frequency __

The method is to use a frequency( * f_{U }*), where only an approximate value is known.

This is used with a known frequency(* f_{K }*) close to the approximate value of

*to produce beats.*

**f**_{U}

The beat frequency ( ** f_{B}** ) is given by:

or (depending on the relative magnitudes of * f_{K }* and

**)**

*f*_{U}

bringing the two equations together,

This is quite an accurate method, achieving results of 0.01% accuracy.

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